nLab
club in a 2-category

Clubs in a 2-category

Idea

We can generalize the notion of club over a cartesian monad on CatCat to a cartesian monad SS on an object CC of any 2-category KK.

Note that the CatCat on which the cartesian monad lives does not generalize to the 2-category KK; it generalizes to the object CC. The 2-category KK generalizes the larger 2-category CATCAT that contains CatCat as an object.

Definition

The context

Let KK be a 2-category and CKC\in K an object that has representable finite limits, meaning that each hom-category K(X,C)K(X,C) has finite limits and the precomposition functors (f):K(X,C)K(Y,C)(-\circ f) : K(X,C) \to K(Y,C) preserve finite limits.

Given morphisms f,g:XCf,g:X\to C, a 2-cell α:fg\alpha:f\to g is called cartesian if for any h,k:YXh,k:Y\to X and 2-cell β:hk\beta:h\to k, the commutative square

fh fβ fk αh αk gh gβ gk \array{ f h & \overset{f \beta}{\to} & f k\\ ^{\alpha h}\downarrow && \downarrow^{\alpha k}\\ g h & \underset{g\beta}{\to} & g k }

is a pullback in K(Y,C)K(Y,C). (Note that this is a different notion of “cartesian 2-cell” than the one involved in the notion of fibration in a 2-category.)

A monad ss on CC (that is, a 1-morphism s:CCs:C\to C equipped with 2-morphisms μ:sss\mu : s s\to s and η:id Cs\eta : id_C \to s that are associative and unital) is cartesian if μ\mu and η\eta are cartesian 2-cells, in the above sense, and ss preserves pullbacks in the sense that each functor (s):K(X,C)K(X,C)(s\circ -):K(X,C) \to K(X,C) preserves pullbacks.

The object of collections

Now suppose that KK is cartesian closed — the weak bicategorical sense suffices, i.e. we have equivalences K(X×Y,Z)K(X,[Y,Z])K(X\times Y,Z) \simeq K(X,[Y,Z]). Suppose further more that KK has a terminal object and comma objects. To avoid confusion, we will denote by 1\mathbf{1} the terminal object of KK, by idid the identity morphisms in KK, and by 1K(X,C)1\in K(X,C) the representable terminal object of CC.

Now any monad ss transposes to a morphism 1[C,C]\mathbf{1}\to [C,C], and we can form the “slice object” [C,C]/s[C,C]/s by the following comma square:

[C,C]/s [C,C] 1 s [C,C] \array{ [C,C]/s & \to & [C,C] \\ \downarrow & \swArrow & \downarrow \\ \mathbf{1} & \underset{s}{\to} & [C,C] }

This has the representable property that K(X,[C,C]/s)K(X×C,C)/s XK(X,[C,C]/s) \simeq K(X\times C,C)/s_X, where s Xs_X is the composite X×Cπ 2CsCX\times C \overset{\pi_2}{\to} C \overset{s}{\to} C.

Similarly, we have s1:1Cs 1 : \mathbf{1} \to C (meaning the composite 11CsC\mathbf{1} \overset{1}{\to} C \overset{s}{\to} C, where 1:1C1:\mathbf{1}\to C is the representable terminal object of CC), and we can form the slice C/s1C/s1. Thus has the representable property K(X,C/s1)K(X,C)/(s1) XK(X,C/s1) \simeq K(X,C)/(s1)_X, where (s1) X(s1)_X is the composite X!1s1CX \overset{!}{\to} \mathbf{1} \overset{s1}{\to} C. Now “evaluate at 1” yields a morphism ev 1:[C,C]/sC/s1ev_1:[C,C]/s \to C/s1, which can be defined representably in a straightforward way.

Given the assumption that CC has representable pullbacks as well, we claim that the morphism ev 1ev_1 has a right adjoint. This can be constructed representably as follows. A morphism XC/s1X\to C/s1 is a morphism a:XCa:X\to C together with a 2-cell a(s1) Xa \to (s1)_X, and we want a morphism b:X×CCb:X\times C\to C together with a 2-cell bs Xb \to s_X. We define bb to be the pullback

b aπ 1 s X (s1) X×C=(s1) Xπ 1 \array{ b & \to & a \pi_1\\ \downarrow && \downarrow\\ s_X & \to & (s1)_{X\times C} = (s1)_X \pi_1 }

It is straightforward to check that this defines a right adjoint whose counit is fully faithful. It commutes with precomposition (since the pullbacks in SS do), so it induces a right adjoint C/s1[C,C]/sC/s1 \to [C,C]/s whose counit is also fully faithful, i.e. C/s1C/s1 is a “reflective subobject” of [C,C]/s[C,C]/s.

Moreover, by construction, a morphism X[C,C]/sX\to [C,C]/s factors through C/s1C/s1 just when for the corresponding map b:X×CCb:X\times C\to C, the induced square

b b(id×1)π 1 s X (s1) Xπ 1 \array{ b & \to & b (id\times 1) \pi_1\\ \downarrow && \downarrow\\ s_X & \to & (s1)_X \pi_1 }

is a pullback. Now for any h,k:YX×Ch,k:Y\to X\times C and β:hk\beta:h\to k, consider the rectangle

bh bk b(id×1)π 1h s Xh s Xk (s1) Yπ 1 \array{b h &\to & b k & \to & b (id \times 1) \pi_1 h\\ \downarrow && \downarrow && \downarrow\\ s_X h & \to & s_X k & \to & (s1)_Y \pi_1 }

The factorization property above says that the right-hand square and the outer rectangle are pullbacks, which by the pasting law for pullbacks implies that the left-hand square is also a pullback, i.e. that bs Xb\to s_X is a cartesian 2-cell as defined above. On the other hand, if bs Xb\to s_X is a cartesian 2-cell, then precomposing with the unique 2-cell id C1id_C \to 1 yields the above square. Hence, C/s1C/s1 as a subobject of [C,C]/s[C,C]/s can be said to be the “full subobject consisting of the cartesian transformations into ss”. We may call it the object of collections over SS.

The substitution product

Now, [C,C][C,C] is always a pseudomonoid in KK, and since ss is a monoid therein, the comma object [C,C]/s[C,C]/s inherits a pseudomonoid structure. We claim that if ss is cartesian, then C/s1C/s1 is closed under this pseudomonoid structure, i.e. C/s1C/s1 is a pseudomonoid and the inclusion is a pseudomonoid morphism.

Firstly, the unit of [C,C]/s[C,C]/s is the map 1[C,C]/s\mathbf{1} \to [C,C]/s corresponding to id Cid_C equipped with η:id Cs\eta:id_C \to s. Since part of ss being cartesian means that η\eta is a cartesian 2-cell, this certainly factors through C/s1C/s1.

Secondly, for the tensor product we again argue representably. Suppose given two maps X[C,C]/sX\to [C,C]/s, corresponding to a,b:X×CCa,b:X\times C\to C equipped with cartetsian 2-cells to s Xs_X. Their “product” induced by the pseudomonoid structure on [C,C]/s[C,C]/s is the composite

X×CΔX×X×Cid×bX×CaC X\times C \overset{\Delta}{\to} X\times X\times C \overset{id\times b}{\to} X\times C \overset{a}{\to} C

equipped with the 2-cell

a(id×b)Δs X(id×b)Δsbssμs. a (id\times b) \Delta \to s_X(id\times b)\Delta \cong s b \to s s \overset{\mu}{\to} s.

Then we have the composite rectangle

a(id×b)Δ s X(id×b)Δ sb ss X s X a(id×b)Δ(id×1)π 1 s X(id×b)Δ(id×1)π 1 sb(id×1)π 1 ss X(id×1)π 1 s1 \array{ a (id\times b) \Delta & \to & s_X(id\times b)\Delta & \cong & s b & \to & s s_X & \to & s_X \\ \downarrow && \downarrow && \downarrow && \downarrow && \downarrow \\ a (id\times b) \Delta (id\times 1) \pi_1 & \to & s_X(id\times b)\Delta(id\times 1)\pi_1 & \cong & s b (id\times 1) \pi_1 & \to & s s_X (id \times 1) \pi_1 & \to & s 1 }

The first square on the left is a pullback since as Xa\to s_X is cartesian. The second is an isomorphism, hence automatically a pullback. The third is a pullback since bs Xb\to s_X is cartesian and ss preserves pullbacks. And the last is a pullback since μ:sss\mu:s s \to s is cartesian. Thus, the composite rectangle is also a pullback, showing that a(id×b)Δs Xa(id\times b)\Delta \to s_X is also cartesian.

Note that we did not actually need CC to have all representable pullbacks nor ss to preserve all of them: we only needed the existence of pullbacks of the form

b aπ 1 s X (s1) Xπ 1 \array{ b & \to & a \pi_1\\ \downarrow && \downarrow\\ s_X & \to & (s1)_X \pi_1 }

and ss to preserve these.

Examples

Clubs on Cat

When K=CATK=CAT, we obtain the notion of club over a cartesian monad described at club. The usual subcase of interest is when C=CatC=Cat, and the simple case of clubs over P\mathbf{P} (the permutation category) is the further special case when ss is the free symmetric (strict) monoidal category monad.

Double clubs

When K=DBLCATK=DBLCAT is the 2-category of (pseudo) double categories, pseudo double functors, and vertical transformations, we obtain essentially the notion of double club from (Garner). Representable pullbacks in DBLCATDBLCAT can be constructed “levelwise”, by taking pullbacks in the category of objects and vertical arrows, and also in the category of (all) horizontal arrows and cells; except that in general such a pullback may not define a pseudo double functor (it may not preserve identities and composition up to isomorphism). The condition (hps) required by Garner ensures that the specific pullbacks needed above do exist as pseudo double functors. Cartesianness of monads and transformations can also be tested levelwise.

Virtual double clubs

When K=VEQUIPK=VEQUIP is the 2-category of virtual equipments (or even just virtual double categories with units) and normal (unit-preserving) functors, we obtain a notion of virtual double club. The units are necessary to have a cartesian closed 2-category (specifically, we need morphisms 1C\mathbf{1} \to C, where 1\mathbf{1} is the terminal object, to correspond to objects of CC; and without units and normality such a morphism would instead be a monoid in CC). However, we can do without requiring composites to exist or be preserved by functors.

This means that it suffices to assume Garner’s condition (hps2) on units, but not necessarily his condition (hps1) on composites. This is nice because in many cases, such as the double category CatCat of categories, functors, and profunctors, (hps2) is automatically true for all monads ss by (Garner, Proposition 48).

On the other hand, in this context we only get that the “inclusion” C/s1[C,C]/sC/s1 \to [C,C]/s is a normal functor of virtual double categories, i.e. lax in the horizontal direction; whereas in Garner’s context it is a strong functor.

References

  • Richard Garner, Double clubs, Cahiers de Topologie et Geometrie Differentielle Categoriques 47 (2006), no. 4, 261–317; arXiv

Created on April 14, 2016 at 19:18:12. See the history of this page for a list of all contributions to it.